Shen's conjecture on groups with given same order type

Abstract

For any group $G$, we define an equivalence relation $\sim$ as below: $$\forall g,h\in Gg\sim h⟺|g|=|h|$$ the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denote by $\alpha (G)$. In this paper, we give a partial answer to a conjecture raised by Shen. In fact, we show that if $G$ is a nilpotent group, then $|\pi (G)|\le |\alpha (G)|$, where $\pi (G)$ is the set of prime divisors of order of $G$. Also we investigate the groups all of whoseproper subgroups, say $H$ have $|\alpha (H)|\le 2$.